Statistical Data Analysis Stat 2: Monte Carlo Method ...cowan/stat/stat_2.pdf · G. Cowan Statistical Data Analysis / Stat 2 13 Monte Carlo detector simulation Takes as input the

G. Cowan Statistical Data Analysis / Stat 2 1

Statistical Data Analysis Stat 2: Monte Carlo Method, Statistical Tests

London Postgraduate Lectures on Particle Physics;

University of London MSci course PH4515

Glen Cowan Physics Department Royal Holloway, University of London [email protected] www.pp.rhul.ac.uk/~cowan

Course web page: www.pp.rhul.ac.uk/~cowan/stat_course.html


What it is: a numerical technique for calculating probabilities and related quantities using sequences of random numbers.

The usual steps:

(1) Generate sequence r1, r2, ..., rm uniform in [0, 1].

(2) Use this to produce another sequence x1, x2, ..., xn distributed according to some pdf f (x) in which we’re interested (x can be a vector).

(3) Use the x values to estimate some property of f (x), e.g., fraction of x values with a < x < b gives

→ MC calculation = integration (at least formally)

MC generated values = ‘simulated data’ → use for testing statistical procedures

The Monte Carlo method


Random number generators Goal: generate uniformly distributed values in [0, 1].

Toss coin for e.g. 32 bit number... (too tiring). → ‘random number generator’

= computer algorithm to generate r1, r2, ..., rn.

Example: multiplicative linear congruential generator (MLCG) ni+1 = (a ni) mod m , where ni = integer a = multiplier m = modulus n0 = seed (initial value)

N.B. mod = modulus (remainder), e.g. 27 mod 5 = 2. This rule produces a sequence of numbers n0, n1, ...


Random number generators (2) The sequence is (unfortunately) periodic!

Example (see Brandt Ch 4): a = 3, m = 7, n0 = 1

← sequence repeats

Choose a, m to obtain long period (maximum = m - 1); m usually close to the largest integer that can represented in the computer.

Only use a subset of a single period of the sequence.


Random number generators (3) are in [0, 1] but are they ‘random’?

Choose a, m so that the ri pass various tests of randomness: uniform distribution in [0, 1], all values independent (no correlations between pairs),

e.g. L’Ecuyer, Commun. ACM 31 (1988) 742 suggests a = 40692 m = 2147483399

Far better generators available, e.g. TRandom3, based on Mersenne twister algorithm, period = 219937 - 1 (a “Mersenne prime”). See F. James, Comp. Phys. Comm. 60 (1990) 111; Brandt Ch. 4


The transformation method Given r1, r2,..., rn uniform in [0, 1], find x1, x2,..., xn that follow f (x) by finding a suitable transformation x (r).

Require:

i.e.

That is, set and solve for x (r).


Example of the transformation method Exponential pdf:

Set and solve for x (r).

→ works too.)


The acceptance-rejection method

Enclose the pdf in a box:

(1) Generate a random number x, uniform in [xmin, xmax], i.e. r1 is uniform in [0,1].

(2) Generate a 2nd independent random number u uniformly distributed between 0 and fmax, i.e. (3) If u < f (x), then accept x. If not, reject x and repeat.


Example with acceptance-rejection method

If dot below curve, use x value in histogram.


Improving efficiency of the acceptance-rejection method

The fraction of accepted points is equal to the fraction of the box’s area under the curve.

For very peaked distributions, this may be very low and thus the algorithm may be slow.

Improve by enclosing the pdf f(x) in a curve C h(x) that conforms to f(x) more closely, where h(x) is a pdf from which we can generate random values and C is a constant.

Generate points uniformly over C h(x).

If point is below f(x), accept x.


Monte Carlo event generators

Simple example: e+e- → µ+µ-

Generate cosθ and φ:

Less simple: ‘event generators’ for a variety of reactions: e+e- → µ+µ-, hadrons, ... pp → hadrons, D-Y, SUSY,...

e.g. PYTHIA, HERWIG, ISAJET...

Output = ‘events’, i.e., for each event we get a list of generated particles and their momentum vectors, types, etc.

12

A simulated event

PYTHIA Monte Carlo pp → gluino-gluino

G. Cowan Statistical Data Analysis / Stat 2


Monte Carlo detector simulation Takes as input the particle list and momenta from generator.

Simulates detector response: multiple Coulomb scattering (generate scattering angle), particle decays (generate lifetime), ionization energy loss (generate Δ), electromagnetic, hadronic showers, production of signals, electronics response, ...

Output = simulated raw data → input to reconstruction software: track finding, fitting, etc.

Predict what you should see at ‘detector level’ given a certain hypothesis for ‘generator level’. Compare with the real data.

Estimate ‘efficiencies’ = #events found / # events generated.

Programming package: GEANT


Hypotheses A hypothesis H specifies the probability for the data, i.e., the outcome of the observation, here symbolically: x.

x could be uni-/multivariate, continuous or discrete.

E.g. write x ~ f (x|H).

x could represent e.g. observation of a single particle, a single event, or an entire “experiment”.

Possible values of x form the sample space S (or “data space”).

Simple (or “point”) hypothesis: f (x|H) completely specified.

Composite hypothesis: H contains unspecified parameter(s).

The probability for x given H is also called the likelihood of the hypothesis, written L(x|H).


Definition of a test Goal is to make some statement based on the observed data x as to the validity of the possible hypotheses.

Consider e.g. a simple hypothesis H0 and alternative H1.

A test of H0 is defined by specifying a critical region W of the data space such that there is no more than some (small) probability α, assuming H0 is correct, to observe the data there, i.e.,

P(x ∈ W | H0 ) ≤ α

If x is observed in the critical region, reject H0.

α is called the size or significance level of the test.

Critical region also called “rejection” region; complement is acceptance region.


Definition of a test (2) But in general there are an infinite number of possible critical regions that give the same significance level α.

So the choice of the critical region for a test of H0 needs to take into account the alternative hypothesis H1.

Roughly speaking, place the critical region where there is a low probability to be found if H0 is true, but high if H1 is true:


Rejecting a hypothesis Note that rejecting H0 is not necessarily equivalent to the statement that we believe it is false and H1 true. In frequentist statistics only associate probability with outcomes of repeatable observations (the data).

In Bayesian statistics, probability of the hypothesis (degree of belief) would be found using Bayes’ theorem:

which depends on the prior probability π(H).

What makes a frequentist test useful is that we can compute the probability to accept/reject a hypothesis assuming that it is true, or assuming some alternative is true.


Type-I, Type-II errors Rejecting the hypothesis H0 when it is true is a Type-I error.

The maximum probability for this is the size of the test:

P(x ∈ W | H0 ) ≤ α

But we might also accept H0 when it is false, and an alternative H1 is true.

This is called a Type-II error, and occurs with probability

P(x ∈ S - W | H1 ) = β

One minus this is called the power of the test with respect to the alternative H1:

Power = 1 - β


Choosing a critical region To construct a test of a hypothesis H0, we can ask what are the relevant alternatives for which one would like to have a high power.

Maximize power wrt H1 = maximize probability to reject H0 if H1 is true.

Often such a test has a high power not only with respect to a specific point alternative but for a class of alternatives. E.g., using a measurement x ~ Gauss (µ, σ) we may test

H0 : µ = µ0 versus the composite alternative H1 : µ > µ0

We get the highest power with respect to any µ > µ0 by taking the critical region x ≥ xc where the cut-off xc is determined by the significance level such that

α = P(x ≥xc|µ0).


Τest of µ = µ0 vs. µ > µ0 with x ~ Gauss(µ,σ)

Standard Gaussian quantile

Standard Gaussian cumulative distribution


Choice of critical region based on power (3)

But we might consider µ < µ0 as well as µ > µ0 to be viable alternatives, and choose the critical region to contain both high and low x (a two-sided test).

New critical region now gives reasonable power for µ < µ0, but less power for µ > µ0 than the original one-sided test.


No such thing as a model-independent test In general we cannot find a single critical region that gives the maximum power for all possible alternatives (no “Uniformly Most Powerful” test).

In HEP we often try to construct a test of

H0 : Standard Model (or “background only”, etc.)

such that we have a well specified “false discovery rate”,

α = Probability to reject H0 if it is true,

and high power with respect to some interesting alternative,

H1 : SUSY, Z′, etc.

But there is no such thing as a “model independent” test. Any statistical test will inevitably have high power with respect to some alternatives and less power with respect to others.


Example setting for statistical tests: the Large Hadron Collider

Counter-rotating proton beams in 27 km circumference ring

pp centre-of-mass energy 14 TeV

Detectors at 4 pp collision points: ATLAS CMS LHCb (b physics) ALICE (heavy ion physics)

general purpose


The ATLAS detector

2100 physicists 37 countries 167 universities/labs

25 m diameter 46 m length 7000 tonnes ~108 electronic channels

Statistical Data Analysis / Stat 2 25

A simulated SUSY event

high pT muons

high pT jets of hadrons

missing transverse energy

p p

G. Cowan


Background events

This event from Standard Model ttbar production also has high pT jets and muons, and some missing transverse energy.

→ can easily mimic a SUSY event.

G. Cowan


For each reaction we consider we will have a hypothesis for the pdf of , e.g.,

Statistical tests (in a particle physics context) Suppose the result of a measurement for an individual event is a collection of numbers

x1 = number of muons,

x2 = mean pT of jets,

x3 = missing energy, ...

follows some n-dimensional joint pdf, which depends on the type of event produced, i.e., was it

etc. E.g. call H0 the background hypothesis (the event type we want to reject); H1 is signal hypothesis (the type we want).

G. Cowan


Selecting events Suppose we have a data sample with two kinds of events, corresponding to hypotheses H0 and H1 and we want to select those of type H1.

Each event is a point in space. What ‘decision boundary’ should we use to accept/reject events as belonging to event types H0 or H1?

accept H1

H0

Perhaps select events with ‘cuts’:

G. Cowan


Other ways to select events Or maybe use some other sort of decision boundary:

accept H1

H0

accept H1

H0

linear or nonlinear

How can we do this in an ‘optimal’ way?

G. Cowan


Test statistics The boundary of the critical region for an n-dimensional data space x = (x1,..., xn) can be defined by an equation of the form

We can work out the pdfs

Decision boundary is now a single ‘cut’ on t, defining the critical region.

So for an n-dimensional problem we have a corresponding 1-d problem.

where t(x1,…, xn) is a scalar test statistic.


Test statistic based on likelihood ratio How can we choose a test’s critical region in an ‘optimal way’?

Neyman-Pearson lemma states:

To get the highest power for a given significance level in a test of H0, (background) versus H1, (signal) the critical region should have

inside the region, and ≤ c outside, where c is a constant chosen to give a test of the desired size.

Equivalently, optimal scalar test statistic is

N.B. any monotonic function of this is leads to the same test. G. Cowan


Classification viewed as a statistical test

Probability to reject H0 if true (type I error):

α = size of test, significance level, false discovery rate

Probability to accept H0 if H1 true (type II error):

1 - β = power of test with respect to H1

Equivalently if e.g. H0 = background event, H1 = signal event, use efficiencies:


Purity / misclassification rate Consider the probability that an event of signal (s) type classified correctly (i.e., the event selection purity),

Use Bayes’ theorem:

Here W is signal region prior probability

posterior probability = signal purity = 1 – signal misclassification rate

Note purity depends on the prior probability for an event to be signal or background as well as on s/b efficiencies.


Neyman-Pearson doesn’t usually help We usually don’t have explicit formulae for the pdfs f (x|s), f (x|b), so for a given x we can’t evaluate the likelihood ratio

Instead we may have Monte Carlo models for signal and background processes, so we can produce simulated data:

generate x ~ f (x|s) → x1,..., xN

generate x ~ f (x|b) → x1,..., xN This gives samples of “training data” with events of known type.

Can be expensive (1 fully simulated LHC event ~ 1 CPU minute).


Approximate LR from histograms Want t(x) = f (x|s)/ f(x|b) for x here

N (x|s) ≈ f (x|s)

N (x|b) ≈ f (x|b)

N(x|s)

N(x|b)

One possibility is to generate MC data and construct histograms for both signal and background. Use (normalized) histogram values to approximate LR:

x

x

Can work well for single variable.


Approximate LR from 2D-histograms Suppose problem has 2 variables. Try using 2-D histograms:

Approximate pdfs using N (x,y|s), N (x,y|b) in corresponding cells. But if we want M bins for each variable, then in n-dimensions we have Mn cells; can’t generate enough training data to populate.

→ Histogram method usually not usable for n > 1 dimension.

signal background


Strategies for multivariate analysis

Neyman-Pearson lemma gives optimal answer, but cannot be used directly, because we usually don’t have f (x|s), f (x|b).

Histogram method with M bins for n variables requires that we estimate Mn parameters (the values of the pdfs in each cell), so this is rarely practical.

A compromise solution is to assume a certain functional form for the test statistic t (x) with fewer parameters; determine them (using MC) to give best separation between signal and background.

Alternatively, try to estimate the probability densities f (x|s) and f (x|b) (with something better than histograms) and use the estimated pdfs to construct an approximate likelihood ratio.


Multivariate methods Many new (and some old) methods:

Fisher discriminant Neural networks Kernel density methods Support Vector Machines Decision trees Boosting Bagging


Resources on multivariate methods

C.M. Bishop, Pattern Recognition and Machine Learning, Springer, 2006

T. Hastie, R. Tibshirani, J. Friedman, The Elements of Statistical Learning, 2nd ed., Springer, 2009

R. Duda, P. Hart, D. Stork, Pattern Classification, 2nd ed., Wiley, 2001 A. Webb, Statistical Pattern Recognition, 2nd ed., Wiley, 2002.

Ilya Narsky and Frank C. Porter, Statistical Analysis Techniques in Particle Physics, Wiley, 2014.

朱永生（著），数据多元分析，科学出版社，北京，2009。


Software Rapidly growing area of development – two important resources: TMVA, Höcker, Stelzer, Tegenfeldt, Voss, Voss, physics/0703039

From tmva.sourceforge.net, also distributed with ROOT Variety of classifiers Good manual, widely used in HEP

scikit-learn Python-based tools for Machine Learning scikit-learn.org

Large user community


Linear test statistic

Suppose there are n input variables: x = (x1,..., xn).

Consider a linear function:

For a given choice of the coefficients w = (w1,..., wn) we will get pdfs f (y|s) and f (y|b) :


Linear test statistic

Fisher: to get large difference between means and small widths for f (y|s) and f (y|b), maximize the difference squared of the expectation values divided by the sum of the variances:

Setting ∂J / ∂wi = 0 gives:

,


The Fisher discriminant

The resulting coefficients wi define a Fisher discriminant.

Coefficients defined up to multiplicative constant; can also add arbitrary offset, i.e., usually define test statistic as

Boundaries of the test’s critical region are surfaces of constant y(x), here linear (hyperplanes):


Fisher discriminant for Gaussian data

Suppose the pdfs of the input variables, f (x|s) and f (x|b), are both multivariate Gaussians with same covariance but different means:

f (x|s) = Gauss(µs, V)

f (x|b) = Gauss(µb, V) Same covariance Vij = cov[xi, xj]

In this case it can be shown that the Fisher discriminant is

i.e., it is a monotonic function of the likelihood ratio and thus leads to the same critical region. So in this case the Fisher discriminant provides an optimal statistical test.







The activation function For activation function h(·) often use logistic sigmoid:



Network architecture Theorem: An MLP with a single hidden layer having a sufficiently large number of nodes can approximate arbitrarily well the optimal decision boundary. Holds for any continuous non-polynomial activation function Leshno, Lin, Pinkus and Schocken (1993) Neural Networks 6, 861-867

However, the number of required nodes may be very large; cannot train well with finite samples of training data.

Recent advances in Deep Neural Networks have shown important advantages in having multiple hidden layers.

For a particle physics application of Deep Learning, see e.g. Baldi, Sadowski and Whiteson, Nature Communications 5 (2014); arXiv:1402.4735.





Overtraining Including more parameters in a classifier makes its decision boundary increasingly flexible, e.g., more nodes/layers for a neural network.

A “flexible” classifier may conform too closely to the training points; the same boundary will not perform well on an independent test data sample (→ “overtraining”).

training sample independent test sample


Monitoring overtraining If we monitor the fraction of misclassified events (or similar, e.g., error function E(w)) for test and training samples, it will usually decrease for both as the boundary is made more flexible:

error rate

flexibility (e.g., number of nodes/layers in MLP)

test sample training sample

optimum at minimum of error rate for test sample

increase in error rate indicates overtraining

G. Cowan Statistical Data Analysis / Stat 2 page 58

Neural network example from LEP II Signal: e+e- → W+W- (often 4 well separated hadron jets) Background: e+e- → qqgg (4 less well separated hadron jets)

← input variables based on jet structure, event shape, ... none by itself gives much separation.

Neural network output:

(Garrido, Juste and Martinez, ALEPH 96-144)






Naive Bayes method First decorrelate x, i.e., find y = Ax, with cov[yi, yj] = V[yi] δij . Pdfs of x and y are then related by

where

If nonlinear features of g(y) not too important, estimate using product of marginal pdfs:

Do separately for the two hypotheses s and b (separate matrices As and Ab and marginal pdfs gs,i, gb,i). Then define test statistic as

Called Naive Bayes classifier. Reduces problem of estimating an n-dimensional pdf to finding n one-dimensional marginal pdfs.


Kernel-based PDE (KDE) Consider d dimensions, N training events, x1, ..., xN, estimate f (x) with

Use e.g. Gaussian kernel:

kernel bandwidth (smoothing parameter)

x where we want to know pdf

x of ith training event


Gaussian KDE in 1-dimension Suppose the pdf (dashed line) below is not known in closed form, but we can generate events that follow it (the red tick marks):

Goal is to find an approximation to the pdf using the generated date values.


Gaussian KDE in 1-dimension (cont.) Place a kernel pdf (here a Gaussian) centred around each generated event weighted by 1/Nevent:


Gaussian KDE in 1-dimension (cont.) The KDE estimate the pdf is given by the sum of all of the Gaussians:


Choice of kernel width The width h of the Gaussians is analogous to the bin width of a histogram. If it is too small, the estimator has noise:


If width of Gaussian kernels too large, structure is washed out:

Choice of kernel width (cont.)


Various strategies can be applied to choose width h of kernel based trade-off between bias and variance (noise).

Adaptive KDE allows width of kernel to vary, e.g., wide where target pdf is low (few events); narrow where pdf is high.

Advantage of KDE: no training!

Disadvantage of KDE: to evaluate we need to sum Nevent terms, so if we have many events this can be slow.

Special treatment required if kernel extends beyond range where pdf defined. Can e.g., renormalize the kernels to unity inside the allowed range; alternatively “mirror” the events about the boundary (contribution from the mirrored events exactly compensates the amount lost outside the boundary).

Software in ROOT: RooKeysPdf (K. Cranmer, CPC 136:198,2001)

KDE discussion


Each event characterized by 3 variables, x, y, z:


Test example (x, y, z)

no cut on z

z < 0.5 z < 0.25

z < 0.75

x

xx

x

y

y

y

y



Particle i.d. in MiniBooNE Detector is a 12-m diameter tank of mineral oil exposed to a beam of neutrinos and viewed by 1520 photomultiplier tubes:

H.J. Yang, MiniBooNE PID, DNP06

Search for νµ to νe oscillations required particle i.d. using information from the PMTs.


Decision trees Out of all the input variables, find the one for which with a single cut gives best improvement in signal purity:

Example by MiniBooNE experiment, B. Roe et al., NIM 543 (2005) 577

where wi. is the weight of the ith event.

Resulting nodes classified as either signal/background.

Iterate until stop criterion reached based on e.g. purity or minimum number of events in a node. The set of cuts defines the decision boundary.


Finding the best single cut The level of separation within a node can, e.g., be quantified by the Gini coefficient, calculated from the (s or b) purity as:

For a cut that splits a set of events a into subsets b and c, one can quantify the improvement in separation by the change in weighted Gini coefficients:

where, e.g.,

Choose e.g. the cut to the maximize Δ; a variant of this scheme can use instead of Gini e.g. the misclassification rate:


Decision trees (2) The terminal nodes (leaves) are classified a signal or background depending on majority vote (or e.g. signal fraction greater than a specified threshold).

This classifies every point in input-variable space as either signal or background, a decision tree classifier, with discriminant function

f(x) = 1 if x in signal region, -1 otherwise

Decision trees tend to be very sensitive to statistical fluctuations in the training sample.

Methods such as boosting can be used to stabilize the tree.



AdaBoost First initialize the training sample T1 using the original

x1,..., xN event data vectors y1,..., yN true class labels (+1 or -1) w1

(1),..., wN(1) event weights

with the weights equal and normalized such that

Then train the classifier f1(x) (e.g., a decision tree) with a method that uses the event weights. Recall for an event at point x,

f1(x) = +1 for x in signal region, -1 in background region

We will define an iterative procedure that gives a series of classifiers f1(x), f2(x),...


Error rate of the kth classifier At the kth iteration the classifier fk(x) has an error rate

where I(X) = 1 if X is true and is zero otherwise.

Next assign a score to the kth classifier based on its error rate,


Updating the event weights The classifier at each iterative step is found from an updated training sample, in which the weight of event i is modified from step k to step k+1 according to

Here Zk is a normalization factor defined such that the sum of the weights over all events is equal to one.

That is, the weight for event i is increased in the k+1 training sample if it was classified incorrectly in step k.

Idea is that next time around the classifier should pay more attention to this event and try to get it right.


Defining the classifier After K boosting iterations, the final classifier is defined as a weighted linear combination of the fk(x),

One can show that the error rate on the training data of the final classifier satisfies the bound

i.e. as long as the εk < ½ (better than random guessing), with enough boosting iterations every event in the training sample will be classified correctly.



Monitoring overtraining

From MiniBooNE example: Performance stable after a few hundred trees.


A simple example (2D) Consider two variables, x1 and x2, and suppose we have formulas for the joint pdfs for both signal (s) and background (b) events (in real problems the formulas are usually not available).

f(x1|x2) ~ Gaussian, different means for s/b, Gaussians have same σ, which depends on x2, f(x2) ~ exponential, same for both s and b, f(x1, x2) = f(x1|x2) f(x2):


Joint and marginal distributions of x1, x2

background

signal

Distribution f(x2) same for s, b.

So does x2 help discriminate between the two event types?


Likelihood ratio for 2D example Neyman-Pearson lemma says best critical region is determined by the likelihood ratio:

Equivalently we can use any monotonic function of this as a test statistic, e.g.,

Boundary of optimal critical region will be curve of constant ln t, and this depends on x2!


Contours of constant MVA output

Exact likelihood ratio Fisher discriminant


Contours of constant MVA output

Multilayer Perceptron 1 hidden layer with 2 nodes

Boosted Decision Tree 200 iterations (AdaBoost)

Training samples: 105 signal and 105 background events


ROC curve

ROC = “receiver operating characteristic” (term from signal processing). Shows (usually) background rejection (1-εb) versus signal efficiency εs. Higher curve is better; usually analysis focused on a small part of the curve.


2D Example: discussion Even though the distribution of x2 is same for signal and background, x1 and x2 are not independent, so using x2 as an input variable helps.

Here we can understand why: high values of x2 correspond to a smaller σ for the Gaussian of x1. So high x2 means that the value of x1 was well measured.

If we don’t consider x2, then all of the x1 measurements are lumped together. Those with large σ (low x2) “pollute” the well measured events with low σ (high x2).

Often in HEP there may be variables that are characteristic of how well measured an event is (region of detector, number of pile-up vertices,...). Including these variables in a multivariate analysis preserves the information carried by the well-measured events, leading to improved performance. In this example we can understand why x2 is useful, even though both signal and background have same pdf for x2.


Summary on multivariate methods Particle physics has used several multivariate methods for many years:

linear (Fisher) discriminant neural networks naive Bayes

and has in recent years started to use a few more:

boosted decision trees support vector machines kernel density estimation k-nearest neighbour

The emphasis is often on controlling systematic uncertainties between the modeled training data and Nature to avoid false discovery.

Although many classifier outputs are "black boxes", a discovery at 5σ significance with a sophisticated (opaque) method will win the competition if backed up by, say, 4σ evidence from a cut-based method.

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Statistical Data Analysis Stat 2: Monte Carlo Method ...cowan/stat/stat_2.pdf · G. Cowan Statistical Data Analysis / Stat 2 13 Monte Carlo detector simulation Takes as input the